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Matter Waves De Broglie's Hypothesis

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Matter Waves De Broglie's Hypothesis Matter Waves de Broglie's Hypothesis Einstein proposed in his photon model of light that E = hf , (7.1) where E is the energy of the photon and f is the frequency of the associated light wave. This can be rewritten as hc E = , (7.2) λ where λ is the wavelength. Now from relativistic dynamics, we know that u pc = , (7.3) c E where p is the momentum of the particle, E is its relativistic energy, and u its velocity. For a photon, u = c, so, pc = E. Using Eq. (7.2), one finds that h p = . (7.4) λ Thus, a photon carries momentum given by Eq. (7.4), a fact amply verified by Compton’s theory and experiments. Now, experiments such as Young’s double-slit interference experiment indicate that light exhibits wavelike behavior. In contrast, the photoelectric effect and Compton scattering convincingly demonstrate the particle nature of light. Louis de Broglie, in his doctorial thesis, proposed that this dual behavior is not limited to light. He argued that every particle (electrons, atoms, etc.) that carries momentum and energy will exhibit wavelike behavior. The frequency and wavelength of the associated wave and the energy and momentum of the particle obey Eqns. (7.1) and (7.4). de Broglie’s argument was based upon the fact that in relativity, matter and energy are equivalent. It follows that material particles (matter) and photons (energy) should exhibit the same fundamental behavior. Eq (7.4) is now called the de Broglie relation. How would one verify such a hypothesis? One could show that a beam of material particles exhibits diffraction or interference. This is precisely what was done by C. Davisson and L. Germer with a beam of electrons.1 Davisson and Germer bombarded a nickel crystal2 with an electron beam. The scattered electrons exhibited a diffraction pattern identical to that predicted by the Bragg law! 3 1 We know electrons exhibit particle-like behavior due to trajectories in bubble chambers. Picture: Pg. 58 Unit Q, Six Ideas 2 Large Ni crystals were created by D&G accidentally when their vacuum system broke and the Ni sample was oxidized. In the process of removing the oxide layer, larger crystals were obtained. Diagram: Diffraction patterns. Krane p. 88-89. 3 G. P. Thomson and A. Reid did similar experiments with thin foils. 1 An important point to note is that the wavelength of the electrons had to be comparable to the distance D between adjacent atoms (lattice parameter) in nickel. For nickel, D = 0.215 nm.4 In the Davisson-Germer experiment the kinetic energy of the electrons was 54 eV. What was the corresponding wavelength? For K = 54 eV, the electrons are decidedly non-relativistic, so 1 2 K = 2 mu . Now, p = mu, so that p = 2Km. (7.5) Inserting the expression for p in Eq. (7.5) into Eq. (7.4) and solving for λ gives hc λ = . (7.6) 2Kmc2 Substituting values yields λ = 0.167 nm. The Bragg condition5 for n = 1 gives Dsinφ = λ, where φ is the angle between the incident and scattered beams. [Note that this is not the standard Bragg formula!] Solving for φ yields 1 ⎛ λ ⎞ 1 ⎛ 0.167 nm ⎞ φ = sin ⎜ ⎟ = sin ⎜ ⎟ = 51°. ⎝ D ⎠ ⎝ 0.215 nm ⎠ A bright spot was seen at this angle by Davisson and Germer. [Show picture. Also PhET simulation.] Note that for a given kinetic energy, a particle with a smaller mass will have a larger wavelength. Wavelike behavior cannot be observed for macroscopic objects because the large masses (~l kg) give rise to exceedingly small wavelengths, due to the extremely small value of Planck’s constant. In the 1960s a double-slit interference experiment was done with electrons for the first time. The unmistakable double-slit pattern, known for centuries to be exhibited by light was now seen to be displayed by electrons as well.6 [Show picture.] So de Broglie’s bold hypothesis about matter waves was verified by experiments. de Broglie even used his theory to explain the quantization of angular momentum in the Bohr model of the atom (incorrectly, as it turned out). Despite these successes, serious questions remained. What is the nature of the wave associated with a material particle? What is the physical meaning of the amplitude of this wave?7 The Wave Function and Its Interpretation To shed some light on these questions, we examine the double-slit interference experiment more carefully. It seems on the face of it that the wave and particle models of light are in conflict in this experiment. The wave model holds that light waves originating from each slit superpose to form an interference pattern. Let us use a detector array to detect the interference pattern. 8 Detectors in the regions of constructive interference will register high intensities, whereas detectors in regions of destructive interference will record low intensities. In the wave picture, 4 See Tipler & Llewellyn, 4th ed. 5 See Tipler & Llewellyn, 4th Ed. for a good discussion 6 Interference effects have been seen for protons, neutrons, atoms, and molecules, including C-60 7 See AJP 70, 200 (2002) 8 Diagram, Double slit interference with individual photons detected, p. 76, Unit Q, Six Ideas 2 the intensity of the light is proportional to the square of the magnitude of the electric field. More precisely, 2 I = ε0c E , (in vacuum) (7.7) where the angle brackets denote a time average. The electric field at every point in space beyond the slits is given by the superposition of the fields from each slit: E = E1 + E2. (7.8) Thus, 2 2 I = ε0c E = ε0c E1 + E2 c E E E E = ε0 ( 1 + 2 )⋅( 1 + 2 ) ⎛ 2 2 ⎞ = ε0c⎜ E1 + E2 + 2 E1 ⋅ E2 ⎟. ⎝ ⎠ The first two terms in parentheses in the equation above are constants. The third term depends on the phase difference between the two wave trains, which is a function of position. This term gives rise to interference. A Newtonian particle model would predict no interference. There would be just two bright regions along straight-line paths from each slit, with overlap in the middle. However, if the intensity of the light used in the experiment is very low, the detectors do register discrete events, which indicate interaction of a single particle with the detector. If the experiment is done with low-intensity light, the same double-slit interference pattern is formed even though there may be only one photon going thought the entire apparatus at a time. Therefore, one can rule out interaction between photons as a mechanism for transmitting information about the pattern. Clearly, the simple Newtonian prediction is incorrect. So how do the particles “know” where to go in order to form an interference pattern? [PhET simulation.] We can get some insight into this seeming paradox by examining how the interference pattern evolves over time. Initially, the photons that are detected seem to be randomly distributed. However, after a statistically large number of photons have been detected, the double-slit pattern emerges. Thus, one cannot predict exactly where a given photon will strike the detector array. However, we can say that on average, more photons will strike the detectors located in the regions of the constructive interference than those detectors in the regions of destructive interference. In other words, the probability that a given photon will strike a given detector is proportional to the intensity of the interference pattern at that detector predicted by the wave model of light. The wave model describes the statistical distribution of measured positions of a large number of identically prepared photons. This interpretation must also be true for “material particles” such as electrons, which undergo double-slit interference. It is within this context that the de Broglie wave is interpreted. The matter wave associated with each quantum particle is described by a wave function Ψ(r,t) . This wave function is in general a complex quantity, with a real and an imaginary part: Ψ(r,t) = Ψreal (r,t) + iΨimag (r,t). (7.9) 3 The imaginary number i = −1. The probability that the particle will be found at position r in a volume dV at time t is proportional to the intensity of the matter wave and the volume dV. Hence, 2 dP = Ψ(r,t) dV , (7.10) where 2 2 2 Ψ(r,t) = Ψreal (r,t) + Ψimag (r,t). (7.11) The square of the modulus of a complex number (and the modulus itself) is a real number, which 2 the probability has to be. From Eq. (7.10), we see that Ψ(r,t) represents the probability density (i.e., probability per unit volume) for finding the particle at position r at time t. Let us cover up one slit in the double-slit experiment. We note that at the position of a given detector, there is a non-zero probability for a photon passing through the open slit to hit that detector. (We are ignoring single-slit diffraction minima here.) If we now cover the open slit and open the covered one, there is again a non-zero probability for a photon passing through the open slit to hit that the same detector. If both slits are open and the detector is at a position of destructive (double-slit) interference, there is zero probability for a photon coming through the slits to hit that detector. We see why the wave function has to be a complex number: Only two complex numbers with non-zero amplitudes can add together to give zero as required by interference.
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